Skip to main content
SpringerLink
Log in

Output Synchronization of Dynamical Networks Having Nodes with Relative-Degree-One Nonlinear Systems

  • Chapter
  • First Online:
Complex Systems

Part of the book series: Studies in Systems, Decision and Control ((SSDC,volume 55))

  • 1444 Accesses

Abstract

In this study, the output synchronization problem in dynamical networks consisting of nodes that have nonlinear dynamic systems with relative degree one has been explored and solved. The system property of passivity has been found to be a useful tool in solving the output synchronization problem in complex dynamical networks. It is well known, however, that not all dynamic systems possess the passivity property. Nonetheless, it has been shown here how to use passivity in order to output synchronize complex dynamical networks. It is also known that if a nonlinear system is weakly minimum phase and has relative degree one, then it is feedback equivalent to a passive system. Thus, the feedback passivation result has been exploited to investigate and solve the output synchronization problem in dynamical networks. The conditions are found, which do not require the assumption of the negative definiteness property of the outer coupling matrix under which output synchronization of dynamical networks having nodes with relative degree one systems is achieved. Furthermore, it has been found that, when all nodes are feedback equivalent to a strictly passive system, the output synchronization criterion is accomplished with less conservative conditions. An illustrative example along with numerical and simulation results is given to demonstrate the effectiveness of the theoretical findings in this study.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 129.00
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 169.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 169.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

References

  1. Watts, D.: Small worlds: the dynamics of networks between order and randomness In: Princeton Studies in Complexity, Princeton University Press. Princeton

    Google Scholar 

  2. Anadriyevsky, B.R., Matveerv, A.S., Fradkov, A.L.: Control and estimation under information constraints: toward a unified theory of control, computation and communications (in Russian). Avtomatika i Telemechanika 71(4), 34–99 (2010)

    Google Scholar 

  3. Dimirovski, G.M., Gough, N.E., Barnett, S.: Categories in systems and control theory. Int. J. Syst. Sci. 8(9), 1081–1090 (1977)

    Article  MATH  Google Scholar 

  4. Mesarovic, M.D., Takahara, Y.: General Systems Theory: Mathematical Foundations. Academic Press, New York (1975)

    Google Scholar 

  5. Siljak, D.D.: Large-Scale Dynamic Systems: Stability and Structure. North-Holland Books, New York (1978)

    MATH  Google Scholar 

  6. Boccaletti, S., Latora, V., Moreno, Y., Chavez, M., Hwang, D.U.: Complex networks: structure and dynamics. Phys. Rep. 424(4–5), 175–308 (2006)

    Article  MathSciNet  Google Scholar 

  7. Jiang, Y., Lozada-Cassou, M., Vinet, A.: Synchronization and symmetry-breaking bifurcations in constructive networks of coupled chaotic oscillators. Phys. Rev. E: Stat., Nonlin, Soft Matter Phys. 68(6), 1–4 (2003)

    Article  Google Scholar 

  8. Newman, M.E.J.: The structure and function of complex networks. SIAM Rev. 45(2), 167–256 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  9. Palla, G., Derenyi, I., Farkas, I., Vicsek, T.: Uncovering the overlapping community structure of complex networks in nature and society. Nature 435(7043), 814–818 (2005)

    Article  Google Scholar 

  10. Strogatz, S.H.: Exploring complex networks. Nature 410(6825), 268–276 (2001)

    Article  Google Scholar 

  11. Siljak, D.D.: Dynamic graphs. Nonlinear Analysis: Hybrid Systems 2, 544–567 (2008)

    MathSciNet  MATH  Google Scholar 

  12. Bertsekas, D., Gallager, R.: Data Networks. Prentice Hall, Englewood Cliffs (1987)

    MATH  Google Scholar 

  13. Dimirovski, G.M.: System-of-systems, Large-scale Systems, Complex Systems or Complex Networks: Where do we stand? In Jorge Pereira, Open Consultation on Complex Systems Engineering towards System-of-Systems, Brussels, 4 July 2011. The EC-INFSO, Brussels, BE, Position Paper II.9 (2011).

    Google Scholar 

  14. Ilic, M., Zaborsky, J.: Dynamics and Control of Large Electric Power Systems. Wiley, New York (2000)

    Google Scholar 

  15. Wang, X.F., Chen, G.R.: Synchronization in scale-free dynamical networks: robustness and fragility. IEEE Trans. Circuits Syst. I: Fundam. Theory Appl. 49(1), 54–62 (2002)

    Google Scholar 

  16. Arenas, A., Daz-Guilera, A., Kurths, J., Moreno, Y., Zhou, C.: Synchronization in complex networks. Phys. Rep. 469(3), 93–153 (2008)

    Article  MathSciNet  Google Scholar 

  17. Das, A., Lewis, F.L.: Distributed adaptive control for synchronization of unknown nonlinear networked systems. Automatica 46(12), 2014–2021 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  18. Lu, J., Chen, G.: A time-varying complex dynamical network model and its controlled synchronization criteria. IEEE Trans. Autom. Control 50(6), 841–846 (2005)

    Article  MathSciNet  Google Scholar 

  19. Olfati-Saber, R., Fax, J.A., Murray, R.M.: Consensus and cooperation in networked multi-agent systems. Proc. IEEE 95(1), 215–233 (2007)

    Article  Google Scholar 

  20. Wang, X.F., Chen, G.: Synchronization in small-world dynamical networks. International J. Bifurcat. Chaos 12(1), 187−192 (2002)

    Google Scholar 

  21. Wang, L., Jing, Y., Kong, Z., Dimirovski, G.M.: Adaptive exponential synchronization of uncertain complex dynamical networks with delay coupling. Neuro-Quantology: Int. J. Neurol. Quantum Phys. 6(4), 397–404 (2008)

    Google Scholar 

  22. Wang, Z., Wang, Y., Liu, Y.: Global synchronization for discrete-time stochastic complex networks with randomly occurred nonlinearities and mixed time delays. IEEE Trans. Neural Netw. 21 (1), 11–25 (2010)

    Google Scholar 

  23. Rodriguez-Angeles, A., Nijmeijer, H.: Mutual synchronization of robots via estimated state feedback: a cooperative approach. IEEE Trans. Control Syst. Technol. 12(4), 542–554 (2004)

    Article  Google Scholar 

  24. DeLellis, P., diBernardo, M., Garofalo, F.: Novel decentralized adaptive strategies for the synchronization of complex networks. Automatica 45(5), 1312–1318 (2009)

    Article  MathSciNet  Google Scholar 

  25. Estrada, E., Gago, S., Caporossi, G.: Design of highly synchronizable and robust networks. Automatica 46(11), 1835–1842 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  26. Wang, Y.W., Wang, H.O., Xiao, J.W., Guan, Z.H.: Synchronization of complex dynamical net-works under recoverable attacks. Automatica 46(1), 197–203 (2010a)

    Google Scholar 

  27. Zhao, J., Hill, D.J., Liu, T.: Synchronization of complex dynamical networks with switching topology: a switched system point of view. Automatica 45(11), 2502–2511 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  28. Zhu, Z.Q., Hu, H.P.: Robust synchronization by time-varying impulsive control. IEEE Trans. Circuits Syst. II Express Briefs 57(9), 735–739 (2010)

    Article  MathSciNet  Google Scholar 

  29. Liu, T., Dimirovski, G.M., Zhao, J.: Exponential synchronization of complex delayed dynamical networks with general topology. Physica A 387(2–3), 643–652 (2008)

    Article  Google Scholar 

  30. Liu, Y., Dimirovski, G.M., Zhao, J.: Output synchronization of dynamical networks having nodes with relative degree one nonlinear systems. In: Kolemisevska-Gugulovska, T.D., Stankovski, M.J. (eds.) Proceedings of COSY-2011 Papers—Special International Conference on Complex Systems: Synergy of Control, Communications and Computing, Ohrid, The Society of ETAI, Skopje, pp. 115−120, 16–20 September 2011 (2011)

    Google Scholar 

  31. Zhong, W.S., Stefanovski, J.D., Dimirovski, G.M., and Zhao, J.: Decentralized control and synchronization of time-varying complex dynamical network. Kybernetika 45(1), 151–167 (2009)

    Google Scholar 

  32. Zhong, W.S., Dimirovski, G.M., Zhao, J.: Decentralized synchronization of an uncertain complex dynamical network. J. Control Theory Appl. 7(3), 225–230 (2009)

    Google Scholar 

  33. Willems, J.C.: Dissipative dynamical systems part I: general theory. Arch. Ration. Mech. Anal. 45(5), 321–351 (1972)

    Article  MathSciNet  MATH  Google Scholar 

  34. Kokotovic, P., Arcak, M.: Constructive nonlinear control: a historical perspective. Automatica 37(5), 637–662 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  35. Haddad, W.M., Chellaboina, V.: Dissipativity theory and stability of feedback interconnections for hybrid dynamical systems. Math. Prob. Eng. 7(4), 299–335 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  36. Zhao, J., Hill, D.J.: Dissipativity theory for switched systems. IEEE Trans. Autom. Control 53(4), 941–953 (2008)

    Article  MathSciNet  Google Scholar 

  37. Chopra, N., Spong, M.W.: Output synchronization of nonlinear systems with time delay in communication. In: Proceedings of the 45th IEEE Conference on Decision and Control, pp. 4986–4992 (2006)

    Google Scholar 

  38. Igarashi, Y., Hatanaka, T., Fujita, M., Spong, M.W.: Passivity-based output synchronization in SE(3). In: Proceedings of the 27th American Control Conference, pp. 723–728 (2008)

    Google Scholar 

  39. Listmann, K.D., Woolsey, C.A.: Output synchronization of systems in chained form. In: Proceedings of the 48th IEEE Conference on Decision and Control CDC2009 held jointly with the 28th Chinese Control Conference, pp. 3341–3346 (2009)

    Google Scholar 

  40. Zhao, J., Hill, D.J., Liu, T.: Passivity-based output synchronization of dynamical networks with non-identical nodes. In: Proceedings of the 49th IEEE Conference on Decision and Control, pp. 7351–7356 (2010)

    Google Scholar 

  41. Chopra, N., Spong, M.W.: Passivity-based control of multi-agent systems In: Advances in Robot Control. Springer, New York (2006)

    Google Scholar 

  42. Yu, H., Antsaklis, P.J.: Passivity-based output synchronization of networked Euler-Lagrange systems subject to nonholonomic constraints. In: Proceedings of the 29th American Control Conference, pp. 208–213 (2010)

    Google Scholar 

  43. Chopra, N., Spong, M.: Output Synchronization of Nonlinear Systems with Relative Degree One Recent Advances in Learning and Control. Springer, Berlin (2008)

    MATH  Google Scholar 

  44. Yao, J., Guan, Z.H., Hill, D.J.: Passivity-based control and synchronization of general complex dynamical networks. Automatica 45(9), 2107–2113 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  45. Zhu, Y., Cheng, D., Hu, X.: Synchronization of a class of networked passive systems with switching topology. In: Proceedings of the 46th IEEE Conference on Decision and Control, pp. 2271–2276 (2007)

    Google Scholar 

  46. Byrnes, C.I., Isidori, A., Willems, J.C.: Passivity, feedback equivalence, and the global stabilization of minimum phase nonlinear systems. IEEE Trans. Autom. Control 36(11), 1228–1240 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  47. Van der Schaft, A.J.: L2-Gain and Passivity Techniques in Nonlinear Control. Springer, Berlin (2000)

    Book  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. State Key Laboratory of Synthetic Automation for Process Industries, Northeastern University, Shenyang, 110819, Liaoning, People’s Republic of China

    Yanyan Liu & Jun Zhao

  2. Faculty of Engineering, Dogus University, Acibadem, 34722, Kadikoy, Istanbul, Turkey

    Georgi Dimirovski

  3. School FEEIT, SS Cyril and Methodius University, 1000, Skopje, Republic of Macedonia

    Georgi Dimirovski

Authors
  1. Yanyan Liu
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Georgi Dimirovski
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Jun Zhao
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Georgi Dimirovski .

Editor information

Editors and Affiliations

  1. School of Electrical Engineering an, St. Cyril and St. Methodius Univers, Skopje, Macedonia

    Georgi M. Dimirovski

Rights and permissions

Reprints and permissions

Copyright information

© 2016 Springer International Publishing Switzerland

About this chapter

Cite this chapter

Liu, Y., Dimirovski, G., Zhao, J. (2016). Output Synchronization of Dynamical Networks Having Nodes with Relative-Degree-One Nonlinear Systems. In: Dimirovski, G. (eds) Complex Systems. Studies in Systems, Decision and Control, vol 55. Springer, Cham. https://doi.org/10.1007/978-3-319-28860-4_3

Download citation

  • DOI: https://doi.org/10.1007/978-3-319-28860-4_3

  • Published:

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-28858-1

  • Online ISBN: 978-3-319-28860-4

  • eBook Packages: EngineeringEngineering (R0)

Publish with us

Policies and ethics

Search

Navigation